Differential methods

In the differential method, which was first suggested by the Dutch physical chemist Jacobus H. van f Hoff (1852-1911) in 1884, the procedure is to determine rates directly by measuring tangents to the experimental concentration-time curves, and to introduce these into the equations in their dfflerentiai forms. 

The theory of the method is as follows. The instantaneous rate of a reaction of the nth order involving only one reacting substance is proportional to the nth power of its concentration, 

A plot of logjo a against log o therefore will give a straight line if the reaction is of simple order the slope will be of the order n. 

There are two different ways, in which this procedure can be applied. One method is to carry out a single run—that is, to allow the reaction to proceed and determine -d at various times. Tangents can then be drawn at different concentrations, as shown 

Schematic-plots of concentration against time, illustrating the use of the differentia method. Tangents are shown drawn at the initial concentration and at the concentrations a, and 32- 

The use of the differential method is very simple in principle, since it is based on the direct utilization of the generation rate (fA). For instance, for first-order kinetics, the component mass balance in a BR becomes 

The use of the method requires numerical differentiation of the CA t curve. For the concentration derivative, the simplest approach is to use the two-point formula 

Multipoint differentiation formulae are presented in many handbooks, for example, in that by Abramowitz and Stegun [ 1 ]. 

FIGURE A10.5 Determination of the generation rate from an experimentally recorded concentration curve. 

The procedure is easily generalized to arbitrary kinetics. The rate expression can be written as 

Taking logarithms of Equation 3.14, we obtain Equation 3.15- another relationship between the rate at any instant, the rate constant, and the instantaneous concentrations of reactants  

If only about the first 1% of a slow reaction is monitored, a plot of the concentration versus time plot is virtually linear, so the initial rate can be measured with little of the uncertainty associated with fitting a tangent to a curve. 

Determination of the rate constant by the differential method is straightforward either of the above-mentioned plots used to determine a and ft gives k from the intercept. 

The mathematical relationships using the differential method to determine a, P and k are linear, and any commercial spreadsheet or scientific software package will have a suitable fitting program. It is desirable that the fitting software gives the standard deviations of the fitted parameters, so that the statistical errors of reaction orders and rate coefficients are known. 

Although not very commonly used (with the exception of the initial rate procedure for slow reactions), the differential method has the advantage that it makes no assumption about what the reaction order might be (note the contrast with the method of integration, Section 3.3.2), and it allows a clear distinction between the order with respect to concentration and order with respect to time. However, the rate constant is obtained from an intercept by this method and will, therefore, have a relatively high associated error. The initial rates method also has the drawback that it may miss the effect of products on the global kinetics of the process. 

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Differential methods - in these techniques the internal grid coordinates are found via the solution of appropriate elliptic, parabolic or hyperbolic partial differential equations. 

Integrating the rate equation is often diffieult for orders greater than 1 or 2. Therefore, the differential method of analysis is used to seareh the form of the rate equation. If a eomplex equation of the type below fits the data, the rate equation is ... 

If the reaetion rate depends on more than one speeies, use the method of exeess eoupled either with the half-life method or the differential method. If the method of exeess is not suitable, an initial rate plot may be eonstrueted by varying the eoneentration of one reaetant while the eoneentrations of the others are held eonstant. This proeess is repeated until the orders of reaetion of eaeh speeies and the speeifie reaetion rate are evaluated. At level 5, the least-squares analysis ean be employed. 

Prepare a plot of reaction rate (-dC /dt) versus f(C ). If the plot is linear and passes through the origin, the rate equation is consistent with the data, otherwise another equation should be tested. Figure 3-17 shows a schematic of the differential method. 

Figure 3-17. Sohematios of the differential method for data analysis.

After the rates have been determined at a series of reactant concentrations, the differential method of testing rate equations is applied. Smith [3] and Carberry [4] have adequately reviewed the designs of heterogeneous catalytic reactors. The following examples review design problems in a plug flow reactor with a homogeneous phase. 

Figure 4 Reaction kinetics plot showing the use of a differential method of rate determination of PP-N6-PP-g-AA ternary blend. Source Ref. 47.

Direct application of the differential equation is perhaps the simplest method of obtaining kinetic parameters from non-isothermal observations. However, the Freeman—Carroll difference—differential method [531] has proved reasonably easy to apply and the treatment has been expanded to cover all functions f(a). The methods are discussed in a sequence similar to that used in Sect. 6.2. 

Differential temperature method. A differential method has been applied to a study of the iodination of acetone, a pseudo-zeroth-order reaction when [(CHj)2CO] [I2].26 It allows the determination of AW to much higher accuracy than otherwise. The reaction rate is expressed mathematically as... 

Determination of Total Monomeric Anthocyanins by pH Differential Method... 

For many situations, a simple total anthocyanin determination is inappropriate because of interference from polymeric anthocyanins, anthocyanin degradation products, or melanoidins from browning reactions. In those cases, the approach has been to measure the absorbance at two different pH values. The differential method measures the absorbance at two pH valnes and rehes on structural transformations of the anthocyanin chromophore as a function of pH. Anthocyanins switch from a saturated bright red-bluish color at pH 1 to colorless at pH 4.5. Conversely, polymeric anthocyanins and others retain their color at pH 4.5. Thus, measurement of anthocyanin samples at pH 1 and 4.5 can remove the interference of other materials that may show absorbance at the A is-max-... 

The pH differential method was described as a fast and convenient assay for the quantitation of monomeric anthocyanins by Giusti (2001). It was approved by the Association of OfQcial Analytical Chemists (AOAC) in 2005 as a standard method to evaluate total monomeric anthocyanin pigment content in fruit juices, beverages, natural colorants, and wines. The degradation index is the ratio between total and monomeric anthocyanins (Table 6.3.1). The content of total anthocyanins can be obtained by the single pH method and the monomeric anthocyanin by the pH differential method. ... 

Quantitation of anthocyanins has become simple and fast since many anthocy-anin standards became commercially available as external standards in the past decade. When the standards are not available, individual anthocyanins or total monomeric anthocyanins can be determined by the use of a generic external standard such as commercial cyanidin-3-glucoside or other compound structurally similar to the analytes of interest. Individual and total peak areas are measured at 520 nm or their and quantified using external standards by which values are typically slightly different from those via the pH differential method. ... 

Lee, J., Durst, R., and Wrolstad, R., AOAC official method 2005.02 total monomeric anthocyanin pigment content of fruit juices, beverages, natural colorants, and wines by the pH differential method, in Official Methods of Analysis of AOAC International, Horowitz, H., Ed., AOAC, Washington, D.C, 2005. 

In the above-described measurement, which we call the absolute method, all pumps have equal speeds (rpm) owing to interconnection to the same drive-shaft. In order to express, if required, a deviation registered for the analyte concentration, one must calibrate with a standard by varying its rpm (B) with respect to that of the titrant (A) a B/A rpm ratio greater than unity means a proportionally lower concentration and vice versa. In general, the absolute method serves to control a sample stream with nearly constant analyte concentration as a sensor one uses not only electroanalytical but often also optical detectors. However, with considerably varying analyte concentrations the differential method is more attractive its principle is that in the set-up in Fig. 5.15 and with the sensor adjusted to a fixed and most sensitive set-point, the rpm of the sample stream (C) is varied with respect to that of the titrant (A) by a feedback control (see Fig. 5.3a) from the sensor via a regulator towards the... 

Differential methods based on differentiation of experimental concentration versus time data in order to obtain the actual rate of reaction. In these approaches one analyzes the data by postulating various functional relations between the rate of reaction and the concentrations of the various species in the reaction mixture and tests these hypotheses using appropriate plots. 

Differential Methods for the Treatment of Reaction Rate Data... 

Since data are almost invariably taken under isothermal conditions to eliminate the temperature dependence of reaction rate constants, one is primarily concerned with determining the concentration dependence of the rate expression [0(Ct)] and the rate constant at the temperature in question. We will now consider two differential methods that can be used in data analysis. 

Differential procedures are illustrated schematically in Figure 3.1. The first diagram indicates how the rate may be determined from concentration versus time data in a constant volume system the second schematic illustrates the method just described. The third diagram indicates the application of our general differential method to this system. 

The following example illustrates the use of the differential method for the analysis of kinetic data. It also exemplifies some of the problems... 

ILLUSTRATION 3.1 USE OF A DIFFERENTIAL METHOD TO DETERMINE A PSEUDO REACTION RATE EXPRESSION FOR THE IODINE CATALYZED BROMINATION OF m-XYLENE... 

Initial Rate Measurements. Another differential method useful in the determination of reaction rate expressions is the initial rate approach. It involves a series of rate measurements at different initial reactant concentrations but restricted to very small conversions of the limiting reagent (5 to 10% or less). This technique differs from those discussed previ-... 

Techniques for the Analysis of Reaction Rate Data that are Suitable for Use with Either Integral or Differential Methods... 

These concentrations may be used in the various integral and differential methods for the analysis of kinetic data that have been described in previous sections. An example of the use of this approach is given in Illustration 3.5. 

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